Small singular regions of spacetime
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911415794663424 |
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| author | Cudek, Franciszek |
| author_facet | Cudek, Franciszek |
| contents | We prove that every open connected region of relativistic spacetime $(M,\textbf{g})$ that encloses a $b$-incomplete half-curve has an open connected subregion that encloses a $b$-incomplete half-curve and is also 'small' in the following sense: it is the image, under the bundle projection map, of some open region in the (connected) orthonormal frame bundle $O^+M$ over that spacetime which is bounded, and whose closure is Cauchy incomplete, with respect to any 'natural' distance function on $O^+M$. As a corollary, it follows that every $b$-incomplete half-curve can be covered by a sequence of singular regions which are images of a sequence of bounded subsets of $O^+M$ whose diameter, with respect to any 'natural' distance function on $O^+M$, tends to zero. We discuss to what extent these results can be interpreted in favour of the claim that singular structure in classical general relativity is 'localizable'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small singular regions of spacetime Cudek, Franciszek General Relativity and Quantum Cosmology 83C75 We prove that every open connected region of relativistic spacetime $(M,\textbf{g})$ that encloses a $b$-incomplete half-curve has an open connected subregion that encloses a $b$-incomplete half-curve and is also 'small' in the following sense: it is the image, under the bundle projection map, of some open region in the (connected) orthonormal frame bundle $O^+M$ over that spacetime which is bounded, and whose closure is Cauchy incomplete, with respect to any 'natural' distance function on $O^+M$. As a corollary, it follows that every $b$-incomplete half-curve can be covered by a sequence of singular regions which are images of a sequence of bounded subsets of $O^+M$ whose diameter, with respect to any 'natural' distance function on $O^+M$, tends to zero. We discuss to what extent these results can be interpreted in favour of the claim that singular structure in classical general relativity is 'localizable'. |
| title | Small singular regions of spacetime |
| topic | General Relativity and Quantum Cosmology 83C75 |
| url | https://arxiv.org/abs/2511.02676 |